We at CatSynth once again join others in recognizing Pi Day today. This time, Google is joining in as well with one of the special “Google Doodles” on their front page:
The image very nicely captures many of the well-known geometric and trigonometric properties of π in an abstract representation of the Google logo.
Of course, I tend to me be more curious about some of the more esoteric properties that interrelate to other parts of mathematics. For example, consider seemingly unrelated Gaussian Integral.
This is the area underneath the Gaussian function which is usually associated with normal distributions probability and statistics. The interrelation of π and e, which we have presented in previous years, is at play again here.
The approximation of the square root of π is 1.77245385…, and like π itself, is a transcendental number. In addition to its appearance in the Gaussian Integral and the Gamma Function (which we also presented on a past Pi Day), it plays an important part of the ancient mathematical problem of Squaring the Circle, that is constructing a square with the same area as a circle using only a compass and a straight edge. Because the square root of π is transcendental, it suggests that Squaring the Circle is in fact impossible. But that probably won’t stop some people from continuing to try.
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We at CatSynth once again, celebrate Pi Day on its three-digit approximation, March 14 (3-14).
We start with some interesting facts about the digits of pi. We presented statistics about the distribution in our 2007 Pi Day post. From super-computing.org, we present some interesting patterns:
01234567890 first occurs at the 53,217,681,704-th digit of pi. 09876543210 first occurs at the 42,321,758,803-th digit of pi. 777777777777 first occurs at the 368,299,898,266-th digit of pi. 666666666666 first occurs at the 1,221,587,715,177-th digit of pi. 271828182845 first occurs at the 1,016,065,419,627-th of digit pi. (that’s e for those who haven’t memorized it) 314159265358 first occurs at the 1,142,905,318,634-th digit of pi.
Last year, we showed the relationship to the Gamma function, and of course to Euler’s identity, which links pi surprisingly closely to the imaginary constant i and the number e. But it is also surprisingly easy to generate pi from simple sequences of integers. Consider the Madhava-Leibniz formula for pi:
Thus one can generate pi from odd integers and simple arithmetic. Another formula only involving perfect squares of integers comes from the Basel problem (named for the town of Basel in Switzerland):
In recognition of Pi Day, the U.S. House of Representatives passed a resolution this week:
And thus the sad history of pi in politics as exemplified by the Indiana Pi Bill of 1897 is put to rest. Now onto erasing the sad history of science and politics in general of the past eight years…